{"paper":{"title":"Homotopy based algorithms for $\\ell_0$-regularized least-squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.NA","authors_text":"Charles Soussen, David Brie, J\\'er\\^ome Idier, Junbo Duan","submitted_at":"2014-01-31T22:26:17Z","abstract_excerpt":"Sparse signal restoration is usually formulated as the minimization of a quadratic cost function $\\|y-Ax\\|_2^2$, where A is a dictionary and x is an unknown sparse vector. It is well-known that imposing an $\\ell_0$ constraint leads to an NP-hard minimization problem. The convex relaxation approach has received considerable attention, where the $\\ell_0$-norm is replaced by the $\\ell_1$-norm. Among the many efficient $\\ell_1$ solvers, the homotopy algorithm minimizes $\\|y-Ax\\|_2^2+\\lambda\\|x\\|_1$ with respect to x for a continuum of $\\lambda$'s. It is inspired by the piecewise regularity of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.4802","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}